British Prime Minister Theresa May could reach a Brexit deal with the opposition Labour Party within days, a leading Conservative Party figure says, following poor local election results.

But even if the party leaderships reach a Brexit compromise, there is no guarantee that it will pass through parliament, which has roundly rejected May’s proposals three times already.

Both major parties have faced voter backlash in local council elections, with the Conservatives losing 1332 seats that were up for re-election and Labour which would typically aim to get hundreds of seats instead losing 81.

Many voters expressed frustration at May’s failure to take Britain out of the European Union almost three years after the country decided to leave in a referendum.

Ruth Davidson, the Conservatives’ leader in Scotland, said a cross-partisan agreement on Brexit was needed before this month’s European elections, or Britain’s major parties would face an even bigger backlash from voters.

“If we thought yesterday’s results were a wake-up call, just wait for the European elections on the 23rd of May,” Davidson told a party conference in Aberdeen on Saturday local time.

Speaking to reporters afterwards, she said there had been progress in the weeks of talks between the Conservatives and Labour to find a Brexit deal that passes parliamentary muster.

“There is a deal that could be done in the next few days … and I really hope we can get to that point,” she said, describing the results as “a kick up the backside”.

Labour leader Jeremy Corbyn said on Friday there was now a huge impetus on every MP to get a Brexit deal done.

In an indication of the hostility May faces from the most pro-Brexit wing of her party, former leader Iain Duncan Smith renewed his call for her to step down soon, calling her a “caretaker prime minister” after the local election losses.

Complicating the picture, the main beneficiaries of the swing against the two major UK parties were the pro-EU Liberal Democrats, who campaigned on a demand for a new referendum, aiming to reverse Brexit.

Foreign minister Jeremy Hunt also saw a “glimmer of hope” that there might be a deal with Labour soon.

But an EU customs union that prevented Britain from striking its own trade deals was not a viable long-term option for the world’s fifth-largest economy, he said.

Earlier on Saturday, Buzzfeed News reported sources saying that May was optimistic about a deal, and that behind closed doors the government had already compromised on a customs union.

“In the last week government ministers and officials presented Labour with a new offer on a customs arrangement that would effectively see the UK remain in the key aspects of a customs union with the EU,” the sources familiar with the talks said.

One source told Buzzfeed “the offer would be tantamount to the government accepting in full Labour’s demands”.

The political editor of the Spectator magazine, which has close links to the Conservatives, said in a column for the Sun newspaper that there had been an agreement to an initial “comprehensive customs arrangement” very like a customs union.